Injective weak solutions in second-gradient nonlinear elasticity

Abstract

Abstract.We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including "mixed" boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.Mathematics Subject Classification. 74B20, 49K20.

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

“…Moreover also the results for second gradient continua proven by Healey et al [36,44] are not applicable here, as the energy of pantographic sheets is not coercive with respect to the highest order derivatives. Before framing the problem in the appropriate energy space we present here some heuristic preliminary considerations.…”

“…Therefore the results by Healey and Chambon [15,36,44] could not be applied directly and there was the appearance of a lack of the coercivity of considered energy. Indeed the second gradient deformation energy for pantographic sheets is not coercive if one considers the standard Sobolev Space, whose norm involves all second order derivatives.…”

Dynamics of charged elastic bodies under diffusion at large strains

“…, which can be satisfied if γ > d/2 − 1 as employed in (29c) to facilitate usage of the results from [14], cf. (40) below.…”

“…It is important that the singularity of the free energy for det F → 0+ is eliminated together with the spurious Lavrentiev phenomenon (cf. [11]) by the used nonsimple-material concept in cooperation with the Healey-Krömer theorem [14]. It is important that this can be done already on the level of Galerkin approximation, so that no other regularization of the singularity of ϕ at det F → 0+ is not needed, cf.…”

“…The singularity of the kernel of the nonlocal term is chosen to guarantee that deformations with finite energy belong to a suitable fractional Sobolev space. Such device has two benefits: first, by as a consequence of a result of T. Healey and S. Kroemer [14], the determinant of the deformation gradient is bounded from below by a positive constant; second, since the deformation is Lipschitz continuous, it is possible to apply the aforementioned change-of-variable theorem, which is intimately related to the area formula [15], to provide an alternative formulation of the equations that govern the scalar potential that carries the information about long-distance interactions. This fact constitutes a novelty with respect to our previous treatment in [27], where self interactions were not accounted for in the dynamic setting.…”